`PQR` is a triangular park with `PQ=PR=200m`. A.T.V. tower stands at the mid-point of `QR`. If the angles of elevation of the top of the tower at `P`, `Q` and `R` are respectively `45^(ulo)`, `30^(ulo)` and `30^(ulo)` then the height of the tower (in m ) is

Options

A. `100sqrt(3)`
B. `50sqrt(2)`
C. `100`
D. `50`

Answer

To find the height of the tower in the triangular park PQR, we will follow these steps:

### Step 1: Understand the Setup
We have a triangular park PQR where PQ = PR = 200 m. The A.T.V. tower is located at the midpoint M of QR. The angles of elevation from points P, Q, and R to the top of the tower (let's denote the height of the tower as H) are given as 45°, 30°, and 30° respectively.

### Step 2: Use Trigonometric Ratios
From point P, we can use the tangent of the angle of elevation to express the height of the tower in terms of the distance from P to M (the midpoint of QR).

Using the angle of elevation from P:
\[ \tan(45^\circ) = \frac{H}{PM} \]
Since \(\tan(45^\circ) = 1\), we have:
\[ H = PM \quad \text{(Equation 1)} \]

### Step 3: Find PM in terms of H
Next, we will find PM using the triangle PQR. Since M is the midpoint of QR, we can denote QM = MR. Let QM = x. Therefore, QR = 2x.

Using the Pythagorean theorem in triangle PQM:
\[ PQ^2 = PM^2 + QM^2 \]
Substituting the known values:
\[ 200^2 = PM^2 + x^2 \]
Thus:
\[ PM^2 = 200^2 - x^2 \quad \text{(Equation 2)} \]

### Step 4: Use the Angle of Elevation from Q
From point Q, we can also express the height of the tower using the angle of elevation of 30°:
\[ \tan(30^\circ) = \frac{H}{QM} \]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), we have:
\[ H = \frac{QM}{\sqrt{3}} \quad \Rightarrow \quad QM = \sqrt{3}H \quad \text{(Equation 3)} \]

### Step 5: Substitute QM in Equation 2
Now substitute QM from Equation 3 into Equation 2:
\[ PM^2 = 200^2 - (\sqrt{3}H)^2 \]
This simplifies to:
\[ PM^2 = 40000 - 3H^2 \quad \text{(Equation 4)} \]

### Step 6: Equate PM from Equations 1 and 4
From Equation 1, we know \(PM = H\). Substitute this into Equation 4:
\[ H^2 = 40000 - 3H^2 \]
Rearranging gives:
\[ H^2 + 3H^2 = 40000 \]
\[ 4H^2 = 40000 \]
\[ H^2 = 10000 \]
Taking the square root:
\[ H = 100 \text{ m} \]

### Final Answer
The height of the tower is \(H = 100 \text{ m}\).